Invariants
| Level: | $60$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.72.1.157 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}11&48\\18&41\end{bmatrix}$, $\begin{bmatrix}17&28\\53&29\end{bmatrix}$, $\begin{bmatrix}21&16\\29&33\end{bmatrix}$, $\begin{bmatrix}37&12\\0&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.36.1.i.1 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $30720$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 36.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} + x y - 2 x w - y z - z w $ |
| $=$ | $4 x^{2} - x y + 4 x z + 2 x w + y^{2} + y z - y w + 4 z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{4} - 9 x^{3} y - 10 x^{3} z + 3 x^{2} y^{2} - 9 x^{2} y z + 3 x^{2} z^{2} + 3 x y^{2} z + \cdots + 4 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3\,\frac{18872510976xz^{8}-176790479040xz^{7}w-982323527040xz^{6}w^{2}-1083614260176xz^{5}w^{3}+184542750576xz^{4}w^{4}+933506846028xz^{3}w^{5}+584248033296xz^{2}w^{6}+156378436461xzw^{7}+16548679665xw^{8}-4210558848y^{2}z^{7}-79177288512y^{2}z^{6}w-102915754848y^{2}z^{5}w^{2}+180501819120y^{2}z^{4}w^{3}+291021206424y^{2}z^{3}w^{4}+133463278692y^{2}z^{2}w^{5}+25390507374y^{2}zw^{6}+3340061925y^{2}w^{7}+10134274752yz^{8}-146730385152yz^{7}w-503799661680yz^{6}w^{2}-127190014416yz^{5}w^{3}+260136073764yz^{4}w^{4}+89654534616yz^{3}w^{5}-17436203577yz^{2}w^{6}-8841827709yzw^{7}-3340061925yw^{8}+11012682496z^{9}-132203098944z^{8}w-502779806208z^{7}w^{2}-194404770672z^{6}w^{3}+530927871696z^{5}w^{4}+750775152996z^{4}w^{5}+501824264040z^{3}w^{6}+173451885975z^{2}w^{7}+25390507374zw^{8}+1461013733w^{9}}{302661120xz^{8}+1365583296xz^{7}w+958628736xz^{6}w^{2}-804320112xz^{5}w^{3}-999059376xz^{4}w^{4}-237924252xz^{3}w^{5}+20605680xz^{2}w^{6}+7461639xzw^{7}+79443xw^{8}+111114624y^{2}z^{7}+90459712y^{2}z^{6}w-313484832y^{2}z^{5}w^{2}-254829360y^{2}z^{4}w^{3}-10325560y^{2}z^{3}w^{4}+18823212y^{2}z^{2}w^{5}+1711242y^{2}zw^{6}-65745y^{2}w^{7}+219766848yz^{8}+641345280yz^{7}w-185242960yz^{6}w^{2}-369715056yz^{5}w^{3}+36783084yz^{4}w^{4}+26863816yz^{3}w^{5}-11638251yz^{2}w^{6}-1631799yzw^{7}+65745yw^{8}+231864576z^{9}+705187392z^{8}w-4655616z^{7}w^{2}-730873424z^{6}w^{3}-647418768z^{5}w^{4}-256276308z^{4}w^{5}-771592z^{3}w^{6}+19179333z^{2}w^{7}+1711242zw^{8}-65745w^{9}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 12.36.1.i.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{4}-9X^{3}Y+3X^{2}Y^{2}-10X^{3}Z-9X^{2}YZ+3XY^{2}Z+3X^{2}Z^{2}+3Y^{2}Z^{2}-4XZ^{3}+4Z^{4} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ | full Jacobian |
| 20.12.0-4.c.1.1 | $20$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.36.1-12.c.1.6 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.36.1-12.c.1.7 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.144.3-12.b.1.3 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bd.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bo.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-12.bq.1.2 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.fk.1.3 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.144.3-60.fm.1.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.fs.1.4 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.144.3-60.fu.1.4 | $60$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 60.360.13-60.p.1.6 | $60$ | $5$ | $5$ | $13$ | $4$ | $1^{12}$ |
| 60.432.13-60.bi.1.12 | $60$ | $6$ | $6$ | $13$ | $0$ | $1^{12}$ |
| 60.720.25-60.ct.1.15 | $60$ | $10$ | $10$ | $25$ | $7$ | $1^{24}$ |
| 120.144.3-24.ca.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.hm.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.kf.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.kt.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nk.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nk.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nl.1.15 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nl.1.18 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.ns.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.ns.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nt.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.nt.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oa.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oa.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.ob.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.ob.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oe.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.oe.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.of.1.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-24.of.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bib.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bip.1.7 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bkf.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bkt.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byq.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byq.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byr.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byr.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byu.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byu.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byv.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.byv.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzg.1.10 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzg.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzh.1.2 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzh.1.31 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzk.1.9 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzk.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzl.1.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.bzl.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 180.216.7-36.k.1.2 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 180.216.7-36.m.1.2 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |